### 2.2 RS model in string theory

Update
There have been interesting developments in realizing the Randall–Sundrum brane-world model in string
theory. A concrete example was found in type IIB supergravity [170]. The bosonic part of the
10-dimensional action is given by (see for example [355])
Here denotes the string metric. We have also defined the combined three-flux, ,
where , and
The term is the action of localized objects, such as branes.
There is a 10-dimensional solution similar to the one in the RS model

where are four-dimensional coordinates and are coordinates on the 6-dimensional compact
manifold . If there are N coincident D3-branes on the manifold, the 10-dimensional spacetime is
described by where is a five-dimensional Einstein manifold. The warp factor is given
by [418]
where is the distance from the D3 branes in the metric and is given by
Here is the string coupling constant, is related to the string mass scale as and
depends on .
Near the D3 branes, the geometry is . At large , the product structure of
the 10-dimensional metric breaks down and ceases to be a good coordinate, and the 6D
extra-dimensional space is not a product of a five sphere and one-dimensional spacetime. This gives a
minimum distance . This cut-off acts as a TeV brane in the RS model. On the other hand,
for , the AdS geometry smoothly glues into a Calabi–Yau compactification. Thus,
the compact manifold plays the role of the Planck brane in the RS model, acting as a cut-off
of the spacetime. The minimum warp factor is determined by the flux associated
with and and thanks to the warping, it is possible to realize a large hierarchy [170]